Model completeness and direct power

The concept of model completeness for a first order theory T was first formulated by A. ROBINSON [6], who proved the basic results about it and gave several examples of model complete theories in 1956. Two of the most prominent examples of model complete theories are those of the theory of algebraically closed fields and the theory of real closed fields. It was thought that it was likely that direct powers preserve model completeness: however, there are some simple counter examples to this coiijecture (see [3]). Because of the sensitivity of model completeness of a theory to il given language, i t is natural to ask whether one can begin with a model complete theory T in a language L and obtain an extension L* of L in such a way that T x T be model complete in L*. One can always form a new language L*, obtained by adding a new relation symbol for each formula of the original language L, and define for any theory T in L a new theory T* in L* which extends T and is model complete with respect to L* (since it admits elimination of quantifiers). Also, in the entire language of generalized product of FEFERMAN-VAUCHT, one authomatically as a theorem gets elimination of quantifiers, and hence, model completeness. We are only aware of these two solutions to this problem in the literature, both being infinite. In this paper, we will effectively construct a finite extension L* of L in which T x T is model complete whenever T is model complete in L. In addition, we will give some results concerning wo-categoricity, oj,-categoricity, and finite axiomatizability of T x T in L*. We also provide an easy proof for preservation of w-categoricity under direct powers, and give some general results regarding model completeness and direct powers.